Question: How can I solve this set of two ordinary differential equations?

Dear All

Hi

I want to solve a set of two ordinary differential equations by means of Maple. I can not unfortunately perform it as a non-numeric method. It is worth mentioning that by the following commands, Maple returns a solution presented as a DESol form, which is not useful for me. Note that functions C2_U0[i](theta) and C2_V0[i](theta) are obtained in terms of  C1_U0[i](theta) and C1_V0[i](theta) by using boundary conditions bc1_U0[i] and bc1_V0[i] at radius r=a. So, I should determine unknown functions C1_U0[i](theta) and C1_V0[i](theta) by using boundary conditions bc2_U0[i] and bc2_V0[i] at radius r=b. It should be noted that f_U0[i](r,theta) and f_V0[i](r,theta) are known for i=1,2,3,...

On the other hands, It is much better if it is possible to decouple two coupled differential equations to two individual differential equations.

 

Please help me to solve the set of  following equations by any method possible.

 

Thanks in advance.

Best wishes.

 

U0[i]:=unapply(f_U0[i](r,theta)+C1_U0[i](theta)+C2_U0[i](theta)*r,r,theta):

 

V0[i]:=unapply(f_V0[i](r,theta)+C1_V0[i](theta)+C2_V0[i](theta)*r,r,theta):

 

bc1_U0[i]:=U0[i](r,theta)=0;

bc2_U0[i]:=subs(r=b,A11(r,theta))*diff(U0[i](r,theta),r)+subs(r=b,A12(r,theta))*(U0[i](r,theta)/r+diff(V0[i](r,theta),theta)/r)=0;

 

bc1_V0[i]:=V0[i](r,theta)=0;

bc2_V0[i]:=(diff(U0[i](r,theta),theta)/r+diff(V0[i](r,theta),r)-V0[i](r,theta)/r)=0;

 

C2_U0[i]:=unapply(solve(subs(r=a,bc1_U0[i]),C2_U0[i](theta)),theta);

C2_V0[i]:=unapply(solve(subs(r=a,bc1_V0[i]),C2_V0[i](theta)),theta);

 

ans_bc[i]:=dsolve({subs(r=b,bc2_U0[i]),subs(r=b,bc2_V0[i])},{C1_U0[i](theta),C1_V0[i](theta)});

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